Slashdot Log In
Poincare Conjecture Proof Completed
Posted by
samzenpus
on Tue Aug 15, 2006 11:39 PM
from the show-your-work dept.
from the show-your-work dept.
Flamerule writes "A New York Times article has finally provided an update on the status of Grigori Perelman's 2003 rough proof of the Poincaré Conjecture. 3 years ago, Perelman published several papers online explaining his idea for proving the conjecture, but after giving lectures at MIT and several other schools (covered on Slashdot) he returned to Russia, where he's remained silent since. Now, mathematicians in the US and elsewhere have finally finished going over his work and have produced several papers, totaling 1000 pages, that give step-by-step, complete proofs of the conjecture. In addition to winning some or all of the $1,000,000 Millennium Prize, Perelman now seems to be the favorite to receive a Fields Medal at the International Mathematics Union meeting next week, but it's not clear that he'll even show up!"
Related Stories
[+]
Poincaré Conjecture May Be Solved 299 comments
Flamerule writes "The New York Times is now reporting that Dr. Grigori (Grisha) Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg, appears to have solved the famous Poincaré Conjecture, one of the Clay Institute's million-dollar Millennium Prize problems. I first noticed a short blurb about this at the MathWorld homepage last week, but Google searches have revealed almost nothing but the date and times of some of his lectures this month, including a packed session at MIT (photos), in which he reportedly presented material that proves the Conjecture. More specifically, the relevant material comes from a paper ("The entropy formula for the Ricci flow and its geometric applications") from last November, and a follow-up that was just released last month."
[+]
2006 Fields Medalists Announced 132 comments
otisaardvark writes "The 2006 Fields medals, awarded every four years and described as the Nobel Prize for Mathematics, have been awarded at the International Congress of Mathematicians. The winners are Grigory Perelman (famous for the ideas underlying the proof of the Poincare and Thurston geometrization conjectures) — who declined the prize, Terence Tao (a child prodigy famous for proving there are arbitrarily long arithmetic progressions of primes, but who works mainly in nonlinear partial differential equations and harmonic analysis), Wendelin Werner (a probabilist working on links with 2D conformal field theories), and Andrei Okounkov (who works on the interface between algebraic geometry and physics)." Yours Truly wrote to mention that Grigory Perelman actually refused his Fields Medalist, on the grounds that he 'doesn't want to be seen as a figurehead'.
[+]
New Yorker on Perelman and Poincaré Controversy 182 comments
b4stard writes "The New Yorker has an interesting article on the recent proof of the Poincaré conjecture and the controversy surrounding it. This is a very nice read, which, among other things, sheds some light on what may have motivated Perelman in refusing to accept the Fields medal." From the article: "The Fields Medal, like the Nobel Prize, grew, in part, out of a desire to elevate science above national animosities. German mathematicians were excluded from the first I.M.U. congress, in 1924, and, though the ban was lifted before the next one, the trauma it caused led, in 1936, to the establishment of the Fields, a prize intended to be 'as purely international and impersonal as possible.'"
[+]
Another Millenium Problem May Have Been Solved 134 comments
S3D writes "After recent verification of the proof of the Poincaré conjecture, another of the Clay Institute's Millenium Problems may have been solved. This new solution is for Navier-Stokes equations under physically reasonable conditions. Navier-Stocks equations describe the motion of fluid substances such as liquids and gases. Penny Smith has posted an Arxiv paper entitled 'Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System' which may prove the existence of such solutions."
[+]
Science's Breakthrough of the Year 92 comments
johkir writes "Last year, evolution was the breakthrough of the year; We found it full of new developments in understanding how new species originate. But we did get a complaint or two that perhaps we were just paying extra attention to the lively political/religious debate that was taking place over the issue, particularly in the United States.
Perish the thought! Our readers can relax this year: Religion and politics are off the table, and n-dimensional geometry is on instead. This year's Breakthrough salutes the work of a lone, publicity-shy Russian mathematician named Grigori Perelman, who was at the Steklov Institute of Mathematics of the Russian Academy of Sciences until 2005. The work is very technical but has received unusual public attention because Perelman appears to have proven the Poincaré Conjecture (Our coverage from earlier this year), a problem in topology whose solution will earn a $1 million prize from the Clay Mathematics Institute. That's only if Perelman survives what's left of a 2-year gauntlet of critical attack required by the Clay rules, but most mathematicians think he will.
There is also a page of runner-ups. Many of which have been covered here on Slashdot."
This discussion has been archived.
No new comments can be posted.
The Fine Print: The following comments are owned by whoever posted them. We are not responsible for them in any way.
Full
Abbreviated
Hidden
Loading... please wait.
Square Pegs in Round Holes (Score:2, Funny)
Re:Square Pegs in Round Holes (Score:2)
Re:Square Pegs in Round Holes (Score:5, Insightful)
Mathematics is not about numbers and problems - it teaches brain to think. Nothing more.
Parent
Okay, so what you're saying is... (Score:5, Funny)
In Soviet Russia, mathematics teaches you.
Parent
Re:Grigori Perelman, please give us a sign! (Score:5, Funny)
The reason they can't find him in Russia is because he's already living in Sweden.
Parent
Re:Grigori Perelman, please give us a sign! (Score:3, Funny)
Re:Grigori Perelman, please give us a sign! (Score:4, Funny)
Parent
Re:Grigori Perelman, please give us a sign! (Score:3, Informative)
Nice bit of jingoistic xenophobia there, but that's about all that's nice about your post.
Gang Tian, who has co-wrote a guide to Perelman's proof, said in 2004: "He certainly has no interest in material things. If he gets the Fields Medal, there is the issue of whether or not he will accept it." He also refused a prize from the European Mathematical Society many years before that.
He is not being threatened, he is simply a pe
Too Many Pages (Score:3, Funny)
Re:Too Many Pages (Score:3, Funny)
Trust me, 99.9999% of the folks will never follow the link if your short blather is at all close to an accurite summary.
Re:Too Many Pages (Score:3, Interesting)
A rabbit is a donut, not a sphere. (Score:4, Insightful)
Re:A rabbit is a donut, not a sphere. (Score:3, Funny)
Chocolate ones
a million, a thousand, roundness (Score:4, Funny)
Re:a million, a thousand, roundness (Score:3, Funny)
who cares Fields medal? (Score:2)
nytimes is more realistic (Score:2, Informative)
http://news.xinhuanet.com/english/2006-06/04/cont
How does this relate to string theory? (Score:2)
Re:How does this relate to string theory? (Score:2)
Re:How does this relate to string theory? (Score:5, Informative)
According to string physicist Lubos Motl [blogspot.com] the proof indeed important to string theory. The proof based on the flow on the manifold (surface), analogous to heat dissipation - Ricci flow [wikipedia.org]. This flow deform metrics (distance between points of the surface). But this process also describe renormalization [wikipedia.org] of worldsheet - how the physics of the worldsheet [wikipedia.org] (surface which string drawing, moving in space and time) change with changing of the observation scale. That is how phisics of string change then the scale of calculation changed.
Parent
Re:How does this relate to string theory? (Score:3, Funny)
IANAA (I am not an acronymist)
Re:How does this relate to string theory? (Score:5, Interesting)
The Ricci Flow [wikipedia.org] was defined by Richard Hamilton in 1981 as a step towards classifying topological compact 3-manifolds. Classifying 3-manifolds would certainly decide The Poincare Conjecture, as it states that all simply connected compact 3-manifolds are homeomorphic to the sphere. This is an important special case: most proofs of the classification of compact 2-manifolds start out by proving the an analogous statement for the 2-sphere. The Ricci Flow is a differential equation which defines how the shape of a manifold changes in time: given an arbitrary manifold M(0), you can apply the differential equation to it to get manifolds M(t) for (some) positive t, which gradually change shape. However, the Ricci Flow is not volume preserving, so you "renormalize" so that M(t) has constant volume.
The Ricci Flow has the useful property that it tends to make manifolds smoother and smoother. For example, if you started out with a lumpy ball, you would eventually get a smooth ball. It was hoped that it could be proved that if the initial manifold was a compact simply connected 3-manifold, then as t increased, the manifold would tend towards a 3-sphere. Unfortunately, while locally solutions to differential equations always exist, they don't necessarily exist for all time, and for some starting manifolds, eventually you would get to a road-block: a t for which M(t) could not be defined. What Perlman (hopefully) showed was that all road-blocks were of certain types, and that a surgery could be formed that would modify the manifold but not it's topological nature, and then you could again apply the Ricci Flow, until the manifold became a sphere.
Note that this method is useful beyond proving the Poincare Conjecture, as it (again, hopefully) describes all road blocks to extending the Ricci Flow, so that the same tools can be applied to any 3-manifold, and not just simply connected ones. In this manner, assuming Perlman made no mistakes (or that any mistakes can be corrected), it is possible to apply the same arguments to prove the Geometrization Conjecture of Thurston, which classifies 3-manifolds.
Parent
Re:How does this relate to string theory? (Score:4, Interesting)
It seems to me at this Ricci Flow differential equation could be quite useful practically. For example, in pattern recognition, if a computer could build a 3d model of an object using multiple vantage points, then simplify the object to one of the handfull of object types described by Perlman using the Ricci Flow, then this simple catagorization might help in the identification of complex objects (e.g. a donut really is a donut, even if it's been heavily frosted).
Do you know if Perlman's technique for handling the singularities will help with the numerical implementation of this process? Or are these issues numerically simple to solve - but only challenging to solve in proof?
Parent
Re:How does this relate to string theory? (Score:3, Interesting)
The tone of the summary is typical (Score:5, Insightful)
I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake? Perhaps he's gone on to other challenges, or he's wrapped up in some research that has his complete attention. Heck, perhaps he just enjoys math for its own sake and doesn't want to deal with all the side-effects of notoriety.
Re:The tone of the summary is typical (Score:5, Insightful)
I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake?
It isn't a shock that he did it for its own sake at all. Look at the thousands of open source programmers. The shock is that he's been given a million dollars and seem uninterested. Linus Torvalds does Linux for its own sake but if someone gave him a million dollars, he'd take it. Even someone who is not materialistic might think: "hmmm. A million dollars might help many Russian orphans or deliver AIDS drugs to Africans or ..." It is strange for a single person to be neither greedy, nor ambitious nor altruistic ... merely obsessed.
Yes, that's strange. It's rare and therefore strange.
Parent
Re:The tone of the summary is typical (Score:4, Insightful)
Where you see value judgments and a jaded reporter, I see a pretty reasonable surprise. I don't see anything in the article where the reporter suggests that Perelman "should" do anything other than what he is. Surprise, and remarking on an unusual behavior, is *not* approbation.
-b
Parent
Re:The tone of the summary is typical (Score:4, Insightful)
Parent
Re:The tone of the summary is typical (Score:5, Insightful)
Parent
Re:The tone of the summary is typical (Score:3, Insightful)
Sadly, yes, doing something for it's own sake rather than for monetary gain is frowned apon
That is not correct. Look at the hoopla around both Gates and Buffett giving way their money. Look at the adoration of Mother Teresa. Look at the army of fans for Linus Torvalds and Richard Stallman.
and sometimes viewed with fear and confusion,
Sure: anything out of the ordinary will engender fear and confusion. There is a difference between suspecting that someone MAY NOT BE altrustic and "frowning upon" the
TFA is well worth reading (Score:2, Funny)
Quite an interesting character, this Perelman, and his proof could turn out to be a real landmark for mathematics.
I liked this bit:
Whatever he's smoking, I want some!
Re:TFA is well worth reading (Score:3, Insightful)
Side note: the Millenium Prize is a cool million. Which is $24 million less than Adam Sandler makes per movie.
Hurray for the free market! The true value for a personal accomplishment has once again been properly determined and awarded!
Re:TFA is well worth reading (Score:3, Insightful)
Want to make a lot of money, do something the generates a lot of money. I can understand your point of view, but get real...
Re:TFA is well worth reading (Score:3, Insightful)
Innovation in math and science generates more money than any movie.
Consider something obviously fundamental to the way we live, like calculus or Fourier transforms.
It is very foolish to think that the direct and immediate monetary rewards a person receives are any real inidcation of the value their work provides to society.
On the contrary... (Score:5, Insightful)
A Scottish physicist two centuries ago sees a strange bump-like waveform in a canal. It persists for over three miles, moving at nearly constant speed along the canal trench. He writes a paper, calling it a soliton wave and two Dutch mathematicians find a nonlinear partial differential equation that describes its motion. The equation, the Korteweg-De Vries Equation, proves fiendishly hard to solve. Finally, the crew working on the hydrogen bomb, finish the job early, so Ulam decides to use ENIAC to help him solve the Korteweg-De Vries Equation. He attains the first analytic solutions, and the study of soliton waves begins in earnest.
How does this earn a quid? Well, solitons model the way that blips of light move down a fiber-optic cable. The military decides that DARPA-net could run on fiber-optic cables, and uses them in building the early internet. Cellular telephone companies begin using fiber-optic cables to pack 100,000 phone conversations into a single pipe in such a way that they all get separated on the other end of the pipe-- one of the great engineering marvels of our time. We owe the modern internet, cell phones, anything that uses fiber-optics, to the solution of the Korteweg-De Vries equation. There was a similar burst of technology earlier in the last century when some closed-form solutions of the Schrödinger Equation were found.
Truth is, when we solve a major math problem like the Poincaré conjecture, billions of dollars of revenue are generated by new technologies that spring into being because of the new scientific understanding that the solution affords us. A thousand Adam Sandlers will not generate the amount of capital that the solution of the Poincaré conjecture will generate, especially considering that Perelman has shown the world that the Millenium Prize Problems are actually solvable.
Parent
Re:TFA is well worth reading (Score:3, Insightful)
Re:TFA is well worth reading (Score:2)
Re:TFA is well worth reading (Score:5, Interesting)
Parent
Recognition = Worry (Score:4, Insightful)
The curse of the gifted is that niggling worry in the back of the mind that if one accepts praise, one may lose his focus, drive or muse, if you will.
name change? (Score:5, Insightful)
Re:name change? (Score:5, Informative)
Things that are proven, are called theorems. They do depend on axioms, but those are defined as true. Sciences about the real world that can't put up axioms (because that'd require ex facto knowledge about the real world), so they can never be conclusively "proven". Hence well call them theories, like theory of gravity, theory of evolution. A few we've called "laws" as well because they have been so extensively tested, but it is not proven in a strict formal sense.
Parent
Has anyone read the actual article? (Score:5, Informative)
He's turned down the money (Score:5, Interesting)
disillusioned with Academia (Score:3, Interesting)
TFA mentions he has distanced himself from others in the Math community because he has become disillusioned. I read into that my own experience, which involved professors trying to hit on me, others trying to get me to write/edit their papers and then taking the credit, others who weave tall tales with just enough truth to fool grant money providers.
One of my colleagues now believes that Science is actually performing a random walk on the landscape of Truth. Occasionally, the walk stumbles over something
two Perelman anecdotes (Score:4, Interesting)
1) I met him at the Mathematical Sciences Research Institute in Berkeley at a workshop sometime around 1994 and he at that point had ridiculously long fingernails and was quite unkempt, even by the quite weak standards applied to research mathematicians. That was a while ago, of course and that was probably one of his first visits to the US. He gave an incomprehensible energetic talk so what most people commented on was his nails.
2) In 2003 or so, during a limited lecture tour about his proof of the Poincare Conjecture, he responded deftly and hilariously to a comment of Misha Gromov in the audience. Gromov is one of the most difficult people to have in a talk- he is a great mathematician with not much patience and has derailed or rerouted talks by many great researchers, who sometimes get quite flustered. I can't remember the exact wording of the exchange, which is too bad since it was precious, but Gromov asked something like "I don't see how that goes, I'd like to see some more details" and Grisha responded with something like "well, yes, you would" and carried on as he had intended.
Re:I remain skeptical (Score:3, Funny)
Re:I remain skeptical (Score:5, Insightful)
Secondly, I would invite you to write down a complete proof of some well-known mathematical fact, the Stone-Weierstrass [wikipedia.org] theorem say. You must prove this from first principles, starting with axiomatic set theory. I would be very surprised if you even managed to finish and even more surprised if the proof came in at under 1000 pages. This highlights what was mentioned by a sibling of mine: mathematics is divided into small steps and you would never dream of trying to prove something all at once.
Thirdly, this is the first ever proof of the Poincare conjecture. It is quite common in mathematics that a nicer proof of a known fact will be found.
Parent
Re:I remain skeptical (Score:4, Funny)
Parent
Re:A question about hypersphere volumes (Score:5, Informative)
The Jacobian, or unit volume if you will, of a hypersphere [wikipedia.org] has a a highest term of sine, or cosine, which grows as you increase dimension. Specifically, for an n dimensional sphere, the highest power of sine or cosine will be sin^(n-2).
Anyway, to answer your question, integrals of sine or cosine to odd powers produce only functions of other sines and cosines. However, integrals of sine or cosine to even powers produce functions of sin(x), cos(x) and x. The x part gives you your pi, but only does so every second dimension, when the highest power is even.
Here's the integrals of (sin(x))^n, for various n
n=0: x
n=1: - cos(x)
n=2: x/2 - sin(2x)/4
n=3: 1/3 * (cos(x))^3 - cos(x)
n=4: (sin(4 x) - 8 sin(2 x) + 12 x)/32
Parent
Re:High Mips, Low I/O (Score:4, Insightful)
Next time you are in a meeting think about this..
Parent